284 CORRESPONDENCE

Among the ‘ first-class ’ records we have : Lympne (346 ft) 19.2 and Manston (1.54 ft) 17.0. For comparison, Croydon (217 ft) gives 18.6.

The useful ‘ coastal ’ stations, north and south, are : Whitstable 1 1.8, Margate 10.5, Ramsgate 10.0, Dover 13.2, Folkestone 14.0, Dungeness 10.0. Of these last, Dover and Folkestone appear to be ‘ good B ’ ; the others grade downwards and broadly compare with Wye.

Inland, Canterbury provides no snow data, and Lympne lies south of the Downs. Apart from Goudhurst (20 mi west of Wye) which at 290 ft gives an average of only 11.0 and need not be dis- cussed, there are only two very short records from the upland towards the North Downs. Sitting- bourne (224 ft) has 6 years’ observations which support a 30-year average of about 14. Throwley (493 ft) has 4 years’ observations which hint at an average of the order of 23. This makes very scanty evidence, but it leads me to think that on the Downs in East Kent a first-class station at say 500 ft could well give an average of the order of 28, that is about four more than the average at Biggin Hill. In respect of the additional incidence of instability showers this would not be out of keeping with the findings at Whipsnade.

I cordially agree with what Mr. Bonacina says about rainy sleet and soft hail; this is why I think we must begin to place our trust in the ‘ first-class ’ airfields. ‘ Soft hail ’ becomes a besetting problem in the reduction of the records kept by some of the more enthusiastic Victorians; one has to go to the temperature records kept on the day in question to try to elucidate some of their reports of snow ’ in summer months. I have long felt it desirable to bring some rational order into our observations of snowfall; we are still beset by absurdly exaggerated accounts from orga- nizations which ought to know better.

Bedford College, University of London, Regent’s Park, N.W. 1. 17 March 19.58.

AIRFLOW OVER MOUNTAINS : INDETERMINACY OF SOLUTION

By G. A. CORBY and J. S. SAWYER

In his contribution to the correspondence columns of the April issue, Scorer (p. 187) discusses the solution of the equation obtained by perturbation methods for (, the vertical displace- ment of a particle at ( x , z ) in the air flowing over a two-dimensional mountain ridge, viz.

For I independent of z, he writes the solution as

{ = A c o s p z + B s i n p z

where p = + (1’ - k2)i As A and B must be harmonic in the x direction with wavelength 2 n / k we can write this as

‘ (3) 5 = (C cos k x + D sin k x ) cos pz + ( E cos k x + F sin k x ) sin p z . where, so far, C, D, E and F are arbitrary functions of k.

For a Fourier component cos kx at the ground ( z - O), the solution Eq. ( 3 ) becomes

5 = cos k x cos pz + ( E cos k x + F sin k x ) sin p z . . (4)

So far there can be no dispute. The difference of opinion arises when we come to assign values to E and F. Part of Scorer’s latest argument amounts to saying as we have no clear-cut second boundary condition to tell us the proper values of E and F , let us exclude all solutions which we are not obliged to include to satisfy the ground boundary condition, i.e., let us put E = F = 0.’ This appears a rather arbitrary line for a mathematician to take. Surely we should seek an additional physical argument or constraint which makes the solution unique and in the treatments of all other authors (including Scorer himself in the past) specific methods have been sought for deciding on the values of E and F.

CORRESPONDENCE 285

If, for convenience, we examine the complex solution of Eq. (I) , we find that the most general solution is

5 z (G&W + He-lW) elkx . ( 5 )

where G and If are complex arbitrary functions of k . As Eq. (1) contains no complex coefficients the real and imaginary parts of the expression ( 5 ) are themselves solutions of Eq. (1).

Writing G and If in terms of their real and imaginary parts, we have

5 = {(GI + i C2) eiW + (HI + i H,) c i W } eilu . --- [ { (G , + HI) cos k x - (G2 + H 2 ) sin k x } cos pz

+ i [{(G2 + H 2 ) cos kx + (G , + HI) sin k x } cos pz

- {(Gz - H 2 ) cos hx -t (GI - H,) sin k x } sin pz]

+ {(GI - If1) cos kx - (C, - H,) sin k x } sin pz] . . (7)

It will be noted that both the r-I and imaginary parts of Eq. (7) have the same form as Eq. (3). If we now put 5 = eilu at z = 0 we obtain G + H = 1, or GI + f& = 1, G, + H 2 = 0.

The real part of Eqs. (6) or (7) then represents the flow over the ground contour To : cos k x and the imaginary part the flow over lo = sin k x . The real part is precisely equivalent to Eq. (4).

If now, as a result of quite separate reasoning about the second boundary condition, we decide that H = 0 in Eq. ( 5 ) then we quite properly write the solution for the flow over the

mountain C0 = cos k x f ( k ) d k as

5 = real part of 1' ei@ eilu f (k) dk s,

The imaginary part is of course the flow over to = sin k x f ( k ) d k . The form of solution

selected has nothing to do with errors in the use of complex notation - i t is quite deliberate. We are not under the mistaken impression that (real part of eiW eik) = (real part of eiw) (real part of e lk ) .

If we avoided complex notation altogether and proceeded from Eq. (4) the same upper- boundary reasoning would lead to the equivalent decision E = 0 , F = - 1. The retention of a term in sin kx could clearly not then be due to including inadvertently part of the solution appropriate to the ' imaginary ' mountain. The lee-wave term contains the factor - sin k x which is picked out from eikx because of the presence of an additional factor ' i.' Would Dr. Scorer maintain that this too is really appropriate to the ' imaginary ' mountain ?

Another point which has a bearing may be mentioned here. As we are only considering the case with I independent of z , there appears no reason why, if we ' freeze ' a streamline and regard it as a new mountain the flow above should change. Either of the solutions eim or e-iw is satisfactory in this respect but if we adopt any mixture of them, then regarding an upper streamline as a new mountain yields a different flow above. In terms of Eq. (4) this means that only E = 0, F = * 1 provide solutions which satisfy this test. This argument cannot be demolished by citing the case of flow containing nodal surfaces because such surfaces can never occur. This is because the vertical wave length 277/p varies with k and no plane within the fluid could be a nodal surface for all the Fourier components ; this is true whatever secondary boundary condition is adopted - even if this is a rigid upper boundary.

Thus, in spite of Scorer's argument, it appears that the only choice open to us is between G = 0 or H = 0 in Eq. (5). In some techniques for making this choice the final result is really determined by deciding on which side of the mountain ridge, windward or lee, it is appropriate for infinite trains of waves to occur. If these are not acceptable on the windward side we are obliged to take H = 0. If, on the other hand, we predetermined the second boundary condition in the manner Scorer now proyoses and then applied these techniques we should reach the embarrassing conclusion that ' lee ' waves occur with equal amplitude on both sides of the mountain ! In other words, it would be quite inconsistent to insert lee waves on the downstream side only whilst at the same time using the second boundary condition now put forward by Scorer.

O s,

Meteorological Office, Air Ministry, Dunstable, Bedfordshire. 14 May 1958.